Connect Two Points in 3D Space with Three Lines
I have two points in 3D space, $P_1$ and $P_4$.
From either point, I have a line extended ($L_1$ intersecting $P_1$ and $L_3$ intersecting $P_4$). I want to join these two lines together with another line, $L_2$.
I'm trying to determine the lengths of Lines $L_1$, $L_2$, and $L_3$, but only the direction of lines $L_1$ and $L_3$ are known and given by the unit vectors $t_1$ and $t_3$ respectively.
The direction of line $L_2$, or the unit vector $t_2$, is unknown.
I'm not all that great at linear algebra, vector geometry, etc. so correct me if I'm wrong here but I'm guessing I still am missing a variable to define this problem?
So let's also say that the angle between $t_1$ and $t_2$ is known as $alpha_1$ OR that the angle between $t_2$ and $t_3$ is known as $alpha_2$.
Does all of these parameters make this problem solvable?
EDIT:
Thinking about this further it looks like the parameters listed above still leave things under-defined (?)
This is a somewhat open ended problem and I can play around with what values are known and unknown. But I'm trying to determine the minimum number required.
Values that can be considered known or known:
$L_1$, $L_2$, $L_3$, $alpha_1$, $alpha_2$
Values that must be considered as "known": $t_1$, $t_3$
linear-algebra vectors 3d
edited Nov 23 at 9:25
Alex Vong
1,282819
asked Nov 23 at 5:27
CLH
32
add a comment |
I have two points in 3D space, $P_1$ and $P_4$.
From either point, I have a line extended ($L_1$ intersecting $P_1$ and $L_3$ intersecting $P_4$). I want to join these two lines together with another line, $L_2$.
I'm trying to determine the lengths of Lines $L_1$, $L_2$, and $L_3$, but only the direction of lines $L_1$ and $L_3$ are known and given by the unit vectors $t_1$ and $t_3$ respectively.
The direction of line $L_2$, or the unit vector $t_2$, is unknown.
I'm not all that great at linear algebra, vector geometry, etc. so correct me if I'm wrong here but I'm guessing I still am missing a variable to define this problem?
So let's also say that the angle between $t_1$ and $t_2$ is known as $alpha_1$ OR that the angle between $t_2$ and $t_3$ is known as $alpha_2$.
Does all of these parameters make this problem solvable?
EDIT:
Thinking about this further it looks like the parameters listed above still leave things under-defined (?)
This is a somewhat open ended problem and I can play around with what values are known and unknown. But I'm trying to determine the minimum number required.
Values that can be considered known or known:
$L_1$, $L_2$, $L_3$, $alpha_1$, $alpha_2$
Values that must be considered as "known": $t_1$, $t_3$
linear-algebra vectors 3d
edited Nov 23 at 9:25
Alex Vong
1,282819
asked Nov 23 at 5:27
CLH
32
where is this question from? is it a part of homework in some class? it's really unclear what you are trying to find and what is given.
– GuySa
Nov 23 at 7:56
How about $P_1$ and $P_4$, are they known?
– Alex Vong
Nov 23 at 9:35
Yes P1 and P4 are known. This is for designing three dimensional trajectories and the way this problem was originally presented to me this was solved through iteration. However I want to try and solve this more or less analytically.
– CLH
Nov 23 at 14:55
For "designing" this trajectory the user will always specify the position of the start and end points and the direction of the lines at those points. Then the user will specify some other combination of variables.
– CLH
Nov 23 at 15:11
add a comment |
I have two points in 3D space, $P_1$ and $P_4$.
From either point, I have a line extended ($L_1$ intersecting $P_1$ and $L_3$ intersecting $P_4$). I want to join these two lines together with another line, $L_2$.
I'm trying to determine the lengths of Lines $L_1$, $L_2$, and $L_3$, but only the direction of lines $L_1$ and $L_3$ are known and given by the unit vectors $t_1$ and $t_3$ respectively.
The direction of line $L_2$, or the unit vector $t_2$, is unknown.
I'm not all that great at linear algebra, vector geometry, etc. so correct me if I'm wrong here but I'm guessing I still am missing a variable to define this problem?
So let's also say that the angle between $t_1$ and $t_2$ is known as $alpha_1$ OR that the angle between $t_2$ and $t_3$ is known as $alpha_2$.
Does all of these parameters make this problem solvable?
EDIT:
Thinking about this further it looks like the parameters listed above still leave things under-defined (?)
This is a somewhat open ended problem and I can play around with what values are known and unknown. But I'm trying to determine the minimum number required.
Values that can be considered known or known:
$L_1$, $L_2$, $L_3$, $alpha_1$, $alpha_2$
Values that must be considered as "known": $t_1$, $t_3$
linear-algebra vectors 3d
edited Nov 23 at 9:25
Alex Vong
1,282819
asked Nov 23 at 5:27
CLH
32
I have two points in 3D space, $P_1$ and $P_4$.
From either point, I have a line extended ($L_1$ intersecting $P_1$ and $L_3$ intersecting $P_4$). I want to join these two lines together with another line, $L_2$.
I'm trying to determine the lengths of Lines $L_1$, $L_2$, and $L_3$, but only the direction of lines $L_1$ and $L_3$ are known and given by the unit vectors $t_1$ and $t_3$ respectively.
The direction of line $L_2$, or the unit vector $t_2$, is unknown.
I'm not all that great at linear algebra, vector geometry, etc. so correct me if I'm wrong here but I'm guessing I still am missing a variable to define this problem?
So let's also say that the angle between $t_1$ and $t_2$ is known as $alpha_1$ OR that the angle between $t_2$ and $t_3$ is known as $alpha_2$.
Does all of these parameters make this problem solvable?
EDIT:
Thinking about this further it looks like the parameters listed above still leave things under-defined (?)
This is a somewhat open ended problem and I can play around with what values are known and unknown. But I'm trying to determine the minimum number required.
Values that can be considered known or known:
$L_1$, $L_2$, $L_3$, $alpha_1$, $alpha_2$
Values that must be considered as "known": $t_1$, $t_3$
linear-algebra vectors 3d
linear-algebra vectors 3d
edited Nov 23 at 9:25
Alex Vong
1,282819
asked Nov 23 at 5:27
CLH
32
edited Nov 23 at 9:25
Alex Vong
1,282819
asked Nov 23 at 5:27
CLH
32
edited Nov 23 at 9:25
Alex Vong
1,282819
edited Nov 23 at 9:25
Alex Vong
1,282819
edited Nov 23 at 9:25
Alex Vong
1,282819
1,282819
asked Nov 23 at 5:27
CLH
32
asked Nov 23 at 5:27
CLH
32
asked Nov 23 at 5:27
CLH
32
32
where is this question from? is it a part of homework in some class? it's really unclear what you are trying to find and what is given.
– GuySa
Nov 23 at 7:56
How about $P_1$ and $P_4$, are they known?
– Alex Vong
Nov 23 at 9:35
Yes P1 and P4 are known. This is for designing three dimensional trajectories and the way this problem was originally presented to me this was solved through iteration. However I want to try and solve this more or less analytically.
– CLH
Nov 23 at 14:55
For "designing" this trajectory the user will always specify the position of the start and end points and the direction of the lines at those points. Then the user will specify some other combination of variables.
– CLH
Nov 23 at 15:11
add a comment |
where is this question from? is it a part of homework in some class? it's really unclear what you are trying to find and what is given.
– GuySa
Nov 23 at 7:56
How about $P_1$ and $P_4$, are they known?
– Alex Vong
Nov 23 at 9:35
Yes P1 and P4 are known. This is for designing three dimensional trajectories and the way this problem was originally presented to me this was solved through iteration. However I want to try and solve this more or less analytically.
– CLH
Nov 23 at 14:55
For "designing" this trajectory the user will always specify the position of the start and end points and the direction of the lines at those points. Then the user will specify some other combination of variables.
– CLH
Nov 23 at 15:11
where is this question from? is it a part of homework in some class? it's really unclear what you are trying to find and what is given.
– GuySa
Nov 23 at 7:56
where is this question from? is it a part of homework in some class? it's really unclear what you are trying to find and what is given.
– GuySa
Nov 23 at 7:56
How about $P_1$ and $P_4$, are they known?
– Alex Vong
Nov 23 at 9:35
How about $P_1$ and $P_4$, are they known?
– Alex Vong
Nov 23 at 9:35
Yes P1 and P4 are known. This is for designing three dimensional trajectories and the way this problem was originally presented to me this was solved through iteration. However I want to try and solve this more or less analytically.
– CLH
Nov 23 at 14:55
Yes P1 and P4 are known. This is for designing three dimensional trajectories and the way this problem was originally presented to me this was solved through iteration. However I want to try and solve this more or less analytically.
– CLH
Nov 23 at 14:55
For "designing" this trajectory the user will always specify the position of the start and end points and the direction of the lines at those points. Then the user will specify some other combination of variables.
– CLH
Nov 23 at 15:11
For "designing" this trajectory the user will always specify the position of the start and end points and the direction of the lines at those points. Then the user will specify some other combination of variables.
– CLH
Nov 23 at 15:11
add a comment |
1 Answer
1
active
oldest
votes
Firstly, let $vec{p_1}, vec{p_2}, vec{p_3}, vec{p_4}$ be the position vector of point $P_1, P_2, P_3, P_4$ respectively. Notice that $P_1$ and $vec{t_1}$ determines line $L_1$ by $vec{r_1}(s) = vec{p_1} + s vec{t_1}$ where $s in mathbb{R}$. Similarly, $P_4$ and $vec{t_3}$ determines line $L_3$ by $vec{r_3}(s) = vec{p_4} + s vec{t_3}$ where $s in mathbb{R}$.
Following the pattern, line $L_2$ can be parameterized as $vec{r_2}(s) = vec{p_2} + s vec{t_2}$ or $vec{r_2}(s) = vec{p_3} + s vec{t_2}$ where $s in mathbb{R}$, which one to use is up to you.
So we need at most $2$ more additional parameters, ${P_2, vec{t_2}}$ or ${P_3, vec{t_2}}$. Finally, you can draw some pictures to convince yourself that adding only one of the parameters $P_2, P_3, vec{t_2}, alpha_1, alpha_2$ doesn't determine $L_2$. So $2$ is the minimal number of additional parameters required.
Please tell me if you need help in calculating lengths from these parameters.
EDIT: In the above, we do not consider $L_1, L_2$ and $L_3$ as parameters, but if we do so, then the minimal number of additional parameters required drops down to $1$, because in the argument above, we have already shown that $L_1$ and $L_3$ are in fact determined, thus we only need $L_2$ as an additional parameter to determine all lines.
answered Nov 23 at 16:20
Alex Vong
1,282819
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010003%2fconnect-two-points-in-3d-space-with-three-lines%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Firstly, let $vec{p_1}, vec{p_2}, vec{p_3}, vec{p_4}$ be the position vector of point $P_1, P_2, P_3, P_4$ respectively. Notice that $P_1$ and $vec{t_1}$ determines line $L_1$ by $vec{r_1}(s) = vec{p_1} + s vec{t_1}$ where $s in mathbb{R}$. Similarly, $P_4$ and $vec{t_3}$ determines line $L_3$ by $vec{r_3}(s) = vec{p_4} + s vec{t_3}$ where $s in mathbb{R}$.
Following the pattern, line $L_2$ can be parameterized as $vec{r_2}(s) = vec{p_2} + s vec{t_2}$ or $vec{r_2}(s) = vec{p_3} + s vec{t_2}$ where $s in mathbb{R}$, which one to use is up to you.
So we need at most $2$ more additional parameters, ${P_2, vec{t_2}}$ or ${P_3, vec{t_2}}$. Finally, you can draw some pictures to convince yourself that adding only one of the parameters $P_2, P_3, vec{t_2}, alpha_1, alpha_2$ doesn't determine $L_2$. So $2$ is the minimal number of additional parameters required.
Please tell me if you need help in calculating lengths from these parameters.
EDIT: In the above, we do not consider $L_1, L_2$ and $L_3$ as parameters, but if we do so, then the minimal number of additional parameters required drops down to $1$, because in the argument above, we have already shown that $L_1$ and $L_3$ are in fact determined, thus we only need $L_2$ as an additional parameter to determine all lines.
answered Nov 23 at 16:20
Alex Vong
1,282819
add a comment |
Firstly, let $vec{p_1}, vec{p_2}, vec{p_3}, vec{p_4}$ be the position vector of point $P_1, P_2, P_3, P_4$ respectively. Notice that $P_1$ and $vec{t_1}$ determines line $L_1$ by $vec{r_1}(s) = vec{p_1} + s vec{t_1}$ where $s in mathbb{R}$. Similarly, $P_4$ and $vec{t_3}$ determines line $L_3$ by $vec{r_3}(s) = vec{p_4} + s vec{t_3}$ where $s in mathbb{R}$.
Following the pattern, line $L_2$ can be parameterized as $vec{r_2}(s) = vec{p_2} + s vec{t_2}$ or $vec{r_2}(s) = vec{p_3} + s vec{t_2}$ where $s in mathbb{R}$, which one to use is up to you.
So we need at most $2$ more additional parameters, ${P_2, vec{t_2}}$ or ${P_3, vec{t_2}}$. Finally, you can draw some pictures to convince yourself that adding only one of the parameters $P_2, P_3, vec{t_2}, alpha_1, alpha_2$ doesn't determine $L_2$. So $2$ is the minimal number of additional parameters required.
Please tell me if you need help in calculating lengths from these parameters.
EDIT: In the above, we do not consider $L_1, L_2$ and $L_3$ as parameters, but if we do so, then the minimal number of additional parameters required drops down to $1$, because in the argument above, we have already shown that $L_1$ and $L_3$ are in fact determined, thus we only need $L_2$ as an additional parameter to determine all lines.
answered Nov 23 at 16:20
Alex Vong
1,282819
add a comment |
Firstly, let $vec{p_1}, vec{p_2}, vec{p_3}, vec{p_4}$ be the position vector of point $P_1, P_2, P_3, P_4$ respectively. Notice that $P_1$ and $vec{t_1}$ determines line $L_1$ by $vec{r_1}(s) = vec{p_1} + s vec{t_1}$ where $s in mathbb{R}$. Similarly, $P_4$ and $vec{t_3}$ determines line $L_3$ by $vec{r_3}(s) = vec{p_4} + s vec{t_3}$ where $s in mathbb{R}$.
Following the pattern, line $L_2$ can be parameterized as $vec{r_2}(s) = vec{p_2} + s vec{t_2}$ or $vec{r_2}(s) = vec{p_3} + s vec{t_2}$ where $s in mathbb{R}$, which one to use is up to you.
So we need at most $2$ more additional parameters, ${P_2, vec{t_2}}$ or ${P_3, vec{t_2}}$. Finally, you can draw some pictures to convince yourself that adding only one of the parameters $P_2, P_3, vec{t_2}, alpha_1, alpha_2$ doesn't determine $L_2$. So $2$ is the minimal number of additional parameters required.
Please tell me if you need help in calculating lengths from these parameters.
EDIT: In the above, we do not consider $L_1, L_2$ and $L_3$ as parameters, but if we do so, then the minimal number of additional parameters required drops down to $1$, because in the argument above, we have already shown that $L_1$ and $L_3$ are in fact determined, thus we only need $L_2$ as an additional parameter to determine all lines.
answered Nov 23 at 16:20
Alex Vong
1,282819
Firstly, let $vec{p_1}, vec{p_2}, vec{p_3}, vec{p_4}$ be the position vector of point $P_1, P_2, P_3, P_4$ respectively. Notice that $P_1$ and $vec{t_1}$ determines line $L_1$ by $vec{r_1}(s) = vec{p_1} + s vec{t_1}$ where $s in mathbb{R}$. Similarly, $P_4$ and $vec{t_3}$ determines line $L_3$ by $vec{r_3}(s) = vec{p_4} + s vec{t_3}$ where $s in mathbb{R}$.
Following the pattern, line $L_2$ can be parameterized as $vec{r_2}(s) = vec{p_2} + s vec{t_2}$ or $vec{r_2}(s) = vec{p_3} + s vec{t_2}$ where $s in mathbb{R}$, which one to use is up to you.
So we need at most $2$ more additional parameters, ${P_2, vec{t_2}}$ or ${P_3, vec{t_2}}$. Finally, you can draw some pictures to convince yourself that adding only one of the parameters $P_2, P_3, vec{t_2}, alpha_1, alpha_2$ doesn't determine $L_2$. So $2$ is the minimal number of additional parameters required.
Please tell me if you need help in calculating lengths from these parameters.
EDIT: In the above, we do not consider $L_1, L_2$ and $L_3$ as parameters, but if we do so, then the minimal number of additional parameters required drops down to $1$, because in the argument above, we have already shown that $L_1$ and $L_3$ are in fact determined, thus we only need $L_2$ as an additional parameter to determine all lines.
answered Nov 23 at 16:20
Alex Vong
1,282819
edited Nov 24 at 10:04
answered Nov 23 at 16:20
Alex Vong
1,282819
answered Nov 23 at 16:20
Alex Vong
1,282819
answered Nov 23 at 16:20
Alex Vong
1,282819
1,282819
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010003%2fconnect-two-points-in-3d-space-with-three-lines%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
where is this question from? is it a part of homework in some class? it's really unclear what you are trying to find and what is given.
– GuySa
Nov 23 at 7:56
How about $P_1$ and $P_4$, are they known?
– Alex Vong
Nov 23 at 9:35
Yes P1 and P4 are known. This is for designing three dimensional trajectories and the way this problem was originally presented to me this was solved through iteration. However I want to try and solve this more or less analytically.
– CLH
Nov 23 at 14:55
For "designing" this trajectory the user will always specify the position of the start and end points and the direction of the lines at those points. Then the user will specify some other combination of variables.
– CLH
Nov 23 at 15:11